Kinetic chemotaxis tumbling kernel determined from macroscopic quantities

TL;DR

It is shown that when given a special design of initial data, the population density, one specific macroscopic quantity as a function of time, contains sufficient information to recover the tumbling kernel and its associated damping coefficient.

Abstract

Chemotaxis is the physical phenomenon that bacteria adjust their motions according to chemical stimulus. A classical model for this phenomenon is a kinetic equation that describes the velocity jump process whose tumbling/transition kernel uniquely determines the effect of chemical stimulus on bacteria. The model has been shown to be an accurate model that matches with bacteria motion qualitatively. For a quantitative modeling, biophysicists and practitioners are also highly interested in determining the explicit value of the tumbling kernel. Due to the experimental limitations, measurements are typically macroscopic in nature. Do macroscopic quantities contain enough information to recover microscopic behavior? In this paper, we give a positive answer. We show that when given a special design of initial data, the population density, one specific macroscopic quantity as a function of time, contains sufficient information to recover the tumbling kernel and its associated damping coefficient. Moreover, we can read off the chemotaxis tumbling kernel using the values of population density directly from this specific experimental design. This theoretical result using kinetic theory sheds light on how practitioners may conduct experiments in laboratories.

Figures (4)

• Figure 1: For a fixed $t_m>0$, the ellipse with focal points $x_i,x_m$ and radius $t_m$ determines all points $x$ with distance $\|x-x_i\|+\|x-x_m\|=t_m$. As $v_i$ is given, the unique tumbling point $x_i+v_i(t_m-\hat{s})$ is the intersection of the half line starting at $x_i$ in direction $v_i$ with this ellipse.
• Figure 2: Considering the situation of a point measurement of $f_0$ at $(x_m,t_m)$ where the initial velocity is prescribed $\phi(x,v) = \tilde{\phi}(x)\delta_{v_i}(v)$, the support of $f_0(\cdot, t_m,v_i)$ equals $\mathop{\mathrm{supp}}\nolimits\{\tilde{\phi}(x_i +v_i t_m)\}\subset B(x_i+v_i t_m,\varepsilon)$, the translated support of $\tilde{\phi}$. When $\varepsilon$ becomes small, at some point $B(x_i+v_i t_m,\varepsilon)$ no longer contains $x_m$, since $x_m\neq x_i+v_i t_m$.
• Figure 3: Geometry and quantities used in the proof, displayed in 2D. In this figure, $x_o = x_i + v_it_m$, the location of the particle assuming it does not tumble, see definition \ref{['eqn:x_o']}. Note that $t_m$ is the length between $x_i$ and $x_o$. The gray area is $x_i+vs$ for $(s,v)\in U$. This is the annulus $A$ in figure \ref{['sfig:annulus']} translated by $x_i$. The red point $\hat{v}\hat{s}$ is not depicted.
• Figure 4: Perturbation of U by $\mathcal{S}$ in 2D. In a first step, $A:=\{vs\mid (v,s)\in U\}$ is displayed. The red dot marks $\hat{v}\hat{s}$ which is bounded away from the boundaries of $A$ by construction. The yellow slice of an annulus is a neighbourhood of $\hat{v}\hat{s}$ that is bounded by the arches of two circles. The image of $\mathcal{S}(U)$ is obtained by shifting each point in $vs\in A$ by $-v_is$. In this picture, the red dot is $a=\mathcal{S}(\hat{s},\hat{v})$. The image of the yellow area is bounded by the same arches of the circles, but the circles were shifted in direction $-v_i$ such that they touch $0$. One can choose the yellow slice of the annulus large enough such that a ball with radius of order $\varepsilon^{\alpha}$ - whose boundary is depicted in green - is contained in the yellow image area.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2: Unique reconstruction of $\sigma$
• Remark 1
• Lemma 1
• Lemma 2
• proof : Proof of Theorem \ref{['thm:unique_sigma']}
• Remark 2
• proof : Proof of Lemma \ref{['lem:f1']}
• Lemma 3
• proof